The Stretch - Length Tradeoff in Geometric Networks: Average Case and Worst Case Study

Consider a network linking the points of a rate-$1$ Poisson point process on the plane. Write \Psi^{\mbox{ave}}(s) for the minimum possible mean length per unit area of such a network, subject to the constraint that the route-length between every pair of points is at most $s$ times the Euclidean distance. We give upper and lower bounds on the function \Psi^{\mbox{ave}}(s), and on the analogous "worst-case" function \Psi^{\mbox{worst}}(s) where the point configuration is arbitrary subject to average density one per unit area. Our bounds are numerically crude, but raise the question of whether there is an exponent $\alpha$ such that each function has $\Psi(s) \asymp (s-1)^{-\alpha}$ as $s \downarrow 1$.Comment: 33 page

Visualization of Geometric Spanner Algorithms

It is easier to understand an algorithm when it can be seen in interactive mode. The current study implemented four algorithms to construct geometric spanners; the path-greedy, gap-greedy, Theta-graph and Yao-graph algorithms. The data structure visualization framework (http://www.cs.usfca.edu/~galles/visualization/) developed by David Galles was used. Two features were added to allow its use in spanner algorithm visualization: support point-based algorithms and export of the output to Ipe drawing software format. The interactive animations in the framework make steps of visualization beautiful and media controls are available to manage the animations. Visualization does not require extensions to be installed on the web browser. It is available at http://cs.yazd.ac.ir/cgalg/AlgsVis/

On a family of strong geometric spanners that admit local routing strategies

We introduce a family of directed geometric graphs, denoted \paz, that depend on two parameters $\lambda$ and $\theta$. For $0\leq \theta<\frac{\pi}{2}$ and ${1/2} < \lambda < 1$, the \paz graph is a strong $t$-spanner, with $t=\frac{1}{(1-\lambda)\cos\theta}$. The out-degree of a node in the \paz graph is at most $\lfloor2\pi/\min(\theta, \arccos\frac{1}{2\lambda})\rfloor$. Moreover, we show that routing can be achieved locally on \paz. Next, we show that all strong $t$-spanners are also $t$-spanners of the unit disk graph. Simulations for various values of the parameters $\lambda$ and $\theta$ indicate that for random point sets, the spanning ratio of \paz is better than the proven theoretical bounds

Theta-3 is connected

In this paper, we show that the $\theta$-graph with three cones is connected. We also provide an alternative proof of the connectivity of the Yao graph with three cones.Comment: 11 pages, to appear in CGT